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Similar to 2018-11-26 Investigation of the band structure of quantum wells based on gapless and narrow-band semiconductors HgTe and InAs
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2018-11-26 Investigation of the band structure of quantum wells based on gapless and narrow-band semiconductors HgTe and InAs
- 1. Investigation of the band structure of quantum wells based on gapless and narrow-band semiconductors HgTe and InAs 126/11/2018 Leonid BOVKUN Marek POTEMSKI Vladimir GAVRILENKO Milan ORLITA Thesis supervisor (LNCMI-G) Thesis supervisor (IPM RAS) Co-supervisor (LNCMI-G) IPM RAS
- 2. Outline 2 1. Introduction 2. Band structure in HgTe/CdHgTe (theory) 3. Experimental details and principles 4. Results and discussion a) Valence band in single quantum wells HgTe/CdHgTe b) Conduction band in asymmetric quantum wells Hg(Cd)Te/CdHgTe c) Double quantum wells HgTe/CdHgTe 5. Conclusions
- 3. 3 Zinc blende lattice: mercury cadmium telluride CdTe HgTe Capper, P., Bulk Growth of Mercury Cadmium Telluride, in Mercury Cadmium Telluride: Growth, Properties and Applications, P. Capper and J. Garland, Editors. 2010, Wiley. CdTe HgTe Band insulator (normal band structure) Gapless material (inverted band structure) s p p s p p Г6 Г8 Г7 Г8 Г6 Г7 Eg=1.6 eV
- 4. 4 Tunable bandgap of CdxHg1-xTe Bulk composition CdTe HgTe CdTe HgTe CdxHg1-xTe Band insulator (normal band structure) Gapless material (massless Kane Fermions) Gapless material (inverted band structure) xc Lawson, W.D., et al. Journal of Physics and Chemistry of Solids, 1959. 9(3): p. 325-329 Harman, T.C., et al., Solid State Communications, 1964. 2(10): p. 305-308. Rogalski, A. Reports on Progress in Physics, 2005. 68(10): p. 2267 Cd0.17Hg0.83Te
- 5. 40 50 60 70 80 90 100 110 120 -100 -80 -60 -40 -20 0 20 40 60 Энергияприk=0,мэВ Ширина квантовой ямы, Å 1-я подзона зоны проводимости 1-я валентная подзона 2-я валентная подзона 3-я валентная подзона E1 H1 H2 H3 QW width, A Energy,meV 1st conduction subband 1st valence subband 2nd valence subband 3rd valence subband Narrow quantum well inherits CdTe band structure 5 s p HgTe CdTeCdTe p s Growth direction Energy d
- 6. 40 50 60 70 80 90 100 110 120 -100 -80 -60 -40 -20 0 20 40 60 Энергияприk=0,мэВ Ширина квантовой ямы, Å 1-я подзона зоны проводимости 1-я валентная подзона 2-я валентная подзона 3-я валентная подзона E1 H1 H2 H3 QW width, A Energy,meV 1st conduction subband 1st valence subband 2nd valence subband 3rd valence subband Width of single QW matters 6 sp sp s p dc s p Energy Growth direction HgTe CdTeCdTe sp sp d
- 7. 40 50 60 70 80 90 100 110 120 -100 -80 -60 -40 -20 0 20 40 60 Энергияприk=0,мэВ Ширина квантовой ямы, Å 1-я подзона зоны проводимости 1-я валентная подзона 2-я валентная подзона 3-я валентная подзона E1 H1 H2 H3 QW width, A Energy,meV 1st conduction subband 1st valence subband 2nd valence subband 3rd valence subband Topological properties of inverted band structure Bernevig, B.A., et al. Science, 2006. 314(5806): p. 1757-61. Konig, M., et al. Science, 2007. 318(5851): p. 766-70. 7 sp bulk sp
- 8. Normal band structure Dirac cone Semimetal Inverted band structure (2D TI) towards bulk 8 Bulk composition CdTe HgTe CdxHg1-xTe Band insulator (normal band structure) Gapless material (massless Kane Fermions) Gapless material (inverted band structure) QW width xc Band structure engineering
- 9. Open gap inverted band structure (3D TI) Weyl system: two pairs of cones at k≠0 9 NostrainCompressiveTensile Gapless material (inverted band structure) Leubner, P., et al., Physical Review Letters, 2016. 117(8): p. 086403 Leubner, P., PhD Thesis, 2016 Band structure engineering Induced strain CdTe HgTe CdxHg1-xTe Gapless material (inverted band structure) Bulk composition
- 10. 10 CdTe HgTe CdxHg1-xTe NostrainCompressiveTensile Band insulator (normal band structure) Gapless material (massless Kane Fermions) Gapless material (inverted band structure) Gapless material (inverted band structure) Inverted band structure with band gap (3D TI) Normal band structure Dirac cone Semimetal Inverted band structure (2D TI) towards bulk Weyl system: two Dirac cones at k≠0 dc dsm Gated structures Doping Double quantum wells 1D-dimensional systems Superlattices Add-ons xc … Induced strain Bulk composition QW width Band structure engineering
- 11. 11 CdTe HgTe CdxHg1-xTe NostrainCompressiveTensile Band insulator (normal band structure) Gapless material (massless Kane Fermions) Gapless material (inverted band structure) Gapless material (inverted band structure) Inverted band structure with band gap (3D TI) Normal band structure Dirac cone Semimetal Inverted band structure (2D TI) towards bulk Weyl system: two Dirac cones at k≠0 dc dsm Gated structures Doping Double quantum wells 1D-dimensional systems Superlattices THz detectors THz sources X-ray detectors xc … Add-ons Induced strain Bulk composition QW width Band structure engineering “band structure on-demand” CdxHg1-xTe can host different band structures depending on composition, QW width and strain
- 12. Outline 12 1. Introduction 2. Band structure in HgTe/CdHgTe (theory) 3. Experimental details 4. Results and discussion a) Valence band in single quantum wells HgTe/CdHgTe b) Conduction band in asymmetric quantum wells Hg(Cd)Te/CdHgTe c) Double quantum wells HgTe/CdHgTe 5. Conclusions
- 13. ee mm H 2 ˆ )ˆ(ˆ 22 kpk p em pk ˆ perturbation em pk ˆ perturbation 8x8 k∙p Hamiltonian Kane, E.O., Band structure of indium antimonide. Journal of Physics and Chemistry of Solids, 1957. 1(4): p. 249-261 Zholudev, M., Terahertz Spectroscopy of HgCdTe / CdHgTe quantum wells, PhD Thesis, 2013
- 14. in nn i i ii i c uCeuCe )()()( 0, )1( 0, )0( , rrr krkr k perturbation perturbation cci , hhhhi , lhlhi , shshi , Novik, E.G., et al., Physical Review B, 2005. 72(3): p. 035321. 8x8 Kane Hamiltonian Г6 Г8 Г7
- 15. 8x8 Hamiltonian with reduced symmetry 𝐻 = 𝐻8∙8 + 𝑉(𝑧) + 𝐻 𝐵𝐼𝐴 + 𝐻𝐼𝐼𝐴 Structure inversion asymmetry of the confinement potential Bulk inversion asymmetry Interface inversion asymmetry 𝑉(𝑧) 𝐻 𝐵𝐼𝐴 𝐻𝐼𝐼𝐴 no inversion center in the crystal Low symmetry of the atoms at an interface Kane k∙p model Electrostatic potential Winkler, R., Springer Tracts in Modern Physics, 2003. 191: p. 69-130. Tarasenko, S.A., et al., Physical Review B, 2015. 91(8): p. 081302(R). Durnev, M.V, et al., Physical Review B, 2016. 93(7): p. 075434.
- 16. 8x8 Hamiltonian with reduced symmetry Is there any experimental evidences? axial model extended model 𝐻 = 𝐻8∙8 + 𝑉(𝑧) + 𝐻 𝐵𝐼𝐴 + 𝐻𝐼𝐼𝐴 no inversion center in the crystal Low symmetry of the atoms at an interface Kane k∙p model Well-known widely used Electrostatic potential
- 17. 0 2 4 6 8 10 -60 -40 -20 0 20 40 60 Energy(meV) Magnetic field (T) 1 0 1 -1 0 -2 Beyond axial model: anticrossing of Landau levels in quantum well with inverted band structure 𝐻 = ⋯ + 𝐻 𝐵𝐼𝐴 Schultz, M., et al., Physical Review B, 1998. 57(23): p. 14772-14775. Zholudev, M.S., et al., JETP Letters, 2015. 100(12): p. 790-794. n = 0 n = -1 n = 1n = 0𝐻 = 𝐻6∙6 symmetrical 12.2 nm vs 8 nm samples
- 18. 0 2 4 6 8 10 -60 -40 -20 0 20 40 60 Energy(meV) Magnetic field (T) 1 0 1 -1 0 -2 𝐻 = ⋯ + 𝐻 𝐵𝐼𝐴 • Orlita, M., et al. Physical Review B, 2011. 83(11): p. 115307. • Zholudev, M.S., et al. JETP Letters, 2015. 100(12): p. 790-794. B (T) Energy(meV) Beyond axial model: anticrossing of Landau levels in quantum well with inverted band structure
- 19. Spin-splitting in the valence band Landwehr, G., et al., Physica E: Low-dimensional Systems and Nanostructures, 2000. 6(1-4): p. 713-717. Ortner, K., et al., Physical Review B, 2002. 66(7). 𝐻 = 𝐻8∙8 + 𝑉(𝑧) 𝐻 = 𝐻8∙8 + 𝑉 𝑧 ? Minkov, G.M., et al., Hole transport and valence-band dispersion law in a HgTe quantum well with a normal energy spectrum. Physical Review B, 2014. 89(16): p. 165311. 𝐻 = 𝐻8∙8 + 𝑉(𝑧) + 𝐻𝐼𝐼𝐴 Minkov, G.M., et al., Valence band energy spectrum of HgTe quantum wells with an inverted band structure. Physical Review B, 2017. 96(3): p. 035310. 𝐻𝐼𝐼𝐴 (tight-binding) Minkov, G.M., et al., Spin-orbit splitting of valence and conduction bands in HgTe quantum wells near the Dirac point. Physical Review B, 2016. 93(15): p. 155304.
- 20. 8x8 Hamiltonian with reduced symmetry 𝐻 = 𝐻8∙8 + 𝑉(𝑧) + 𝐻 𝐵𝐼𝐴 + 𝐻𝐼𝐼𝐴 no inversion center in the crystal Low symmetry of the atoms at an interface Kane k∙p model Electrostatic potential extended modelaxial model gated/doped samples with induced V(z) EF in the valence band narrow heterostructures Observed and treated separately
- 21. Motivation 𝐻 = 𝐻8∙8 + 𝑉(𝑧) + 𝐻 𝐵𝐼𝐴 + 𝐻𝐼𝐼𝐴 Conduct experimental studies of the band structure in samples with different asymmetry effects Interpret experimental result in the framework of extended model Specify the impact of particular effects When asymmetry effects are relevant?
- 22. Outline 22 1. Introduction 2. Band structure in HgTe/CdHgTe (theory) 3. Experimental details and principles Narrow-gap systems Optical response in FIR range Transparent in far-infrared Magneto-optical experiments 1 2 Band structure ?
- 23. Cyclotron resonance 𝜔𝑐 = 𝑞𝐵 𝑚 Dresselhaus, G., A.F. Kip, and C. Kittel, Cyclotron Resonance of Electrons and Holes in Silicon and Germanium Crystals. Physical Review, 1955. 98(2): p. 368-384. 𝑚( 𝑑 𝑣 𝑑𝑡 + 1 𝜏 𝑣) = 𝑞𝐸 + 𝑞 𝑣 × 𝐵 For a plane-polarized radiation Ex we have 𝜎 = 𝜎0( 1+𝑖𝜔𝜏 1+ 𝜔 𝑐 2−𝜔2 𝜏2+2𝑖𝜔𝜏 ) Required condition for CR: 𝜔𝑐 𝜏 > 1 Powerabsorption
- 24. B = 0 24 Landau levels in a single 2D parabolic band 𝐸 𝒑 = 𝑝 𝑥 2 + 𝑝 𝑦 2 2𝑚 𝐸 𝑛 = ℏ𝜔𝑐(𝑛 + 1 2 ) B ≠ 0 n = 3 n = 2 n = 1 n = 0 electric dipolar selection rules
- 25. CR & interband transitions Band structure Landau levels conduction band valence band Energy(meV) α nm-1 B (T)
- 26. CR & interband transitions Band structure Landau levels conduction band valence band Energy(meV) nm-1 α β α- γ B (T)
- 27. Landau level degeneracy Landau levels β γ EF x 𝐷 = 𝑒𝐵 ℎ x B (T) For each Landau level states available
- 28. Landau levels β γ EF x x B (T) Magneto-optical spectroscopy • intraband transition (CR) • interband transitions m* Eg occupation effect Landau levels Band structure
- 29. 29 Fourier-spectroscopy in the magnetic field
- 30. 30 Fourier-spectroscopy in the magnetic field
- 32. Outline 32 1. Introduction 2. Band structure in HgTe/CdHgTe (theory) 3. Experimental details 4. Results and discussion a) Valence band in single quantum wells HgTe/CdHgTe b) Conduction band in asymmetric quantum wells Hg(Cd)Te/CdHgTe c) Double quantum wells HgTe/CdHgTe 5. Conclusions
- 33. 33 Studies of the valence band normal band structure Gapless Dirac cone inverted band structure Mikhailov, N.N., et al., Int. J. of Nanotechnology, 2006. 3(1): p. 120. Dvoretsky, S., et al. J. of Electronic Materials, 2010. 39(7): p. 918-923. Samples • • • • • 4.6 nm 5.0 nm 5.5 nm 6.0 nm 8.0 nm x=0 y=0.7 y=0.7
- 34. Transport in magnetic fields 34 dQW = 6 nm normal band structure dQW = 5.5 nm normal band structure
- 35. α– and β– is characteristic for p-type only 35 opaque regions dQW = 6 nm (near gapless ) α− 𝐻 = 𝐻8∙8
- 36. p-type samples with different width 36 𝐻 = 𝐻8∙8
- 37. Additional lines observed in low fields 37 𝐻 = 𝐻8∙8
- 38. 38 Additional lines (5 nm QW)
- 39. 39 Beyond axial model: . Mixing of states in 5.0 nm QW β- α- 𝐻 = 𝐻8∙8 + 𝑉(𝑧) + 𝐻 𝐵𝐼𝐴 + 𝐻𝐼𝐼𝐴
- 40. 40 axial model extended model experiment Beyond axial model: anticrossing of Landau levels in quantum well with inverted band structure +𝐻 𝐵𝐼𝐴 + 𝐻𝐼𝐼𝐴
- 41. 41 Beyond axial model: anticrossing of Landau levels both in n-type and p-type samples n-type p-type α β- ββ α- 𝐻𝑒𝑥𝑡
- 42. 42 • Observed lines α– and β– are characteristic for p-type QW only (expected in the axial model) • Beyond axial model: forbidden transitions at low B avoided crossing in p-type • Extended model with reduced symmetry is capable to explain experimental results Studies of the valence band
- 43. Outline 43 1. Introduction 2. Band structure in HgTe/CdHgTe (theory) 3. Experimental details 4. Results and discussion a) Valence band in single quantum wells HgTe/CdHgTe b) Conduction band in asymmetric quantum wells Hg(Cd)Te/CdHgTe c) Double quantum wells HgTe/CdHgTe 5. Conclusions
- 44. 44 Studies of the spin-splitting in conduction band dc In-doped undoped normal band structure inverted band structure Sample A Sample B Samples
- 45. Splitting is different for s and p-states s: ( ) s p p: ( )𝐻 = 𝐻8∙8 + 𝑉(𝑧) Schultz, M., et al., Semiconductor Science and Technology, 1996. 11(8): p. 1168. Zhang, X.C., et al. Physical Review B, 2001. 63(24): p. 245305. Gui, Y.S., et al., Physical Review B, 2004. 70(11): p. 115328. Hinz, J., et al., Semiconductor Science and Technology, 2006. 21(4): p. 501. dQW = 21 nm (symmetrical doping + top gate)
- 46. No spin-splitting in normal band? s p Splitting in conduction band is enhanced due to s-p mixing inverted normal s sp sp s p s ? Winkler, R., Rashba spin splitting in two-dimensional electron and hole systems. Physical Review B, 2000. 62(7): p. 4245-4248. 𝐻 = 𝐻8∙8 + 𝑉(𝑧)
- 47. Carrier density in different subbands 4.2K Sample B (normal) 𝐸1 + 𝐸1 −
- 48. Sample m* 1 m* 2 А Experiment 0,0376 0,0417 Calculations 0,0397 0,0414 В Experiment 0,0459 0,0498 Calculations 0,0448 0,0463 CR line splitting in quasi-classical regime 𝐴 = 𝜋(𝑘 𝐹)2
- 49. Transitions at high magnetic fields
- 50. 8x8 Hamiltonian with reduced symmetry Electrostatic potential 𝐻 = 𝐻8∙8 + 𝑉(𝑧) + 𝐻 𝐵𝐼𝐴 + 𝐻𝐼𝐼𝐴 no inversion center in the crystal Low symmetry of the atoms at an interface 𝑉 𝑧 𝑖𝑠 𝑟𝑒𝑐𝑜𝑛𝑠𝑡𝑟𝑢𝑐𝑡𝑒𝑑 𝑓𝑟𝑜𝑚 𝑡𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑛𝑡
- 51. Calculated band splitting in sample B (normal) s4−s3 s3 s1 s2−s1 EF s2 𝐻𝑒𝑥𝑡 s4 Why splitting (s4-s3) and (s2-s1) is different?
- 52. Splitting in sample B (normal) s34 sp1 EF sp2 sp2 sp1 Splitting in conduction band is enhanced due to s-p mixing inverted normal s sp sp s p s normal s sp sp 𝐻𝑒𝑥𝑡
- 53. 53 Studies of the spin-splitting in conduction band of asymmetrically doped single quantum wells • Electrostatic potential induces spin-splitting even in the sample with normal band structure, due to significant admixture of the hole-like states. • Spin-splitting due to bulk and interface asymmetry was estimated to be more pronounced in second conduction band, but was not resolved experimentally.
- 54. Outline 54 1. Introduction 2. Band structure in HgTe/CdHgTe (theory) 3. Experimental details 4. Results and discussion a) Valence band in single quantum wells HgTe/CdHgTe b) Conduction band in asymmetric quantum wells Hg(Cd)Te/CdHgTe c) Double quantum wells HgTe/CdHgTe 5. Conclusions
- 55. 55 Double quantum well hosts additional phase Krishtopenko, S.S., W. Knap, and F. Teppe, Phase transitions in two tunnel-coupled HgTe quantum wells: Bilayer graphene analogy and beyond. Scientific Reports, 2016. 6: p. 30755. BI band insulator TI topological insulator BG bilayer graphene SM semimetal band structure on demand i.e. “phases” d, nm t, nm A 4.5 (3.8) 3.0 (3.7) B 6.5 (6.3) 3.0 (2.8) Samples:
- 56. Bonding and anti-bonding bands anti- bonding bonding Sample A: dQW = 4.5 (3.8) nm, t = 3.7 nm
- 57. 57 Illumination effect in DQW reveal e-h asymmetry holes electrons Energy(meV) No illumination Short illumination Constant illumination 𝑚ℎ > 𝑚 𝑒 𝑚ℎ 𝑚 𝑒 1.01 0.97 TB/T0 𝑚 𝑒 = 0.0125 𝑚0 𝑚ℎ = 0.0347 𝑚0
- 58. 58 Barrier transparency is different for electrons and holes e h e h 𝑚ℎ𝑜𝑙𝑒𝑠 > 𝑚 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 light heavy 𝑚 𝑒 = 0.0125 𝑚0 𝑚ℎ = 0.0347 𝑚0
- 59. 59 Interband transitions are doubled β− δ−
- 60. 60 Adequate interpretation in axial model 𝐻 = 𝐻8∙8 β− δ−
- 61. 61 Studies of double quantum wells In sample (A) with normal dispersion: • Fermi level could be modified with illumination, revealing band asymmetry • Energy splitting (E2-E1) > (H2-H1) due to the different transparency of the barrier for Г6 and Г8 states • Doubling of interband transitions observed
- 62. Tunnel-coupled Dirac cones anti- bonding bonding Sample B: dQW = 6.5 (6.3) nm, t = 3.0 (2.8) nm 62
- 63. 63 Response of “bilayer graphene”-like sample β1 – intraband (HH1), β2 – interband (E2-HH1), interlayer coupling (𝛾1 ≈ 𝛽2 – 𝛽1) ≈ 20 meV 𝛾1 𝐻 = 𝐻8∙8
- 64. 64 Comparison with bilayer graphene McCann, E. and M. Koshino, Reports on Progress in Physics, 2013. 76(5): p. 056503. γ1 ≈ 20 meV Г-point γ1 ≈ 380 meV K-point
- 65. 65 Magnetic field induces decoupling quadratic E(k) linear E(k) ℏ𝜔 𝑐 ∝ B ℏ𝜔 𝑐 ∝ 𝐵 γ1
- 66. 66 Fundamental gap opening DQW is more sensitive to electrostatic potential than SQW! 𝐻 = 𝐻8∙8 + 𝑉(𝑧) A comparison of electron states for zero and finite electrostatic potential
- 67. Double quantum wells HgTe/CdHgTe the main absorption lines are doubled (vs single well), it indicates tunnel transparency, which is different for s and p-states. Band structure of coupled QW with critical width is analogous to the structure of a bilayer graphene with much lower value of interlayer coupling. Conduction band in asymmetric quantum wells Hg(Cd)Te/CdHgTe Bychkov-Rashba splitting is observed not only in samples with inverted but also with normal band structure. It is due to admixture of the hole-like states in a multicomponent wave function. Valence band in symmetric quantum wells HgTe/CdHgTe Interband transitions between Landau levels that are forbidden within the framework of the widely used axial dispersion law model are observed. It is due to anisotropy of chemical bonds at heterointerfaces and the absence of an inversion center in the crystal lattice. Conclusions: when asymmetry effects are relevant? 67 𝑉 𝑧 ! 𝐻 𝐵𝐼𝐴? 𝐻𝐼𝐼𝐴? 𝐻 𝐵𝐼𝐴 + 𝐻𝐼𝐼𝐴 𝑉 𝑧 !
- 68. Acknowledgements Laboratoire National des Champs Magnétiques Intenses Grenoble, France M. Potemski M. Orlita I. Crassee B. A. Piot C. Faugeras M. Hakl A. Slobodeniuk I. Lobanova M. Molas Y. Henni M. Koperski I. Breslavetz R. Pankow E. Yildiz C. Warth-Martin Université Grenoble Alpes Grenoble, France S. Ferrari J. Collot Institute for Physics of Microstructures Nizhny Novgorod, Russia V.I. Gavrilenko V.Ya. Aleshkin A.V. Ikonnikov S.S. Krishtopenko S.V. Morozov K.V. Maremyanin K.E. Spirin A.M. Kadykov M.S. Zholudev V. Rumyantsev D. Kozlov A. Antonov P. Bushuikin M. Fadeev Laboratoire Charles Coulomb Montpellier, France M. Marcinkiewicz S. Ruffenach C. Consejo F. Teppe W. Knap Rzhanov Institute of Semiconductor Physics Novosibirsk, Russia N.N. Mikhailov S.A. Dvoretskii B.R. Semyagin M.A. Putyato E.A. Emelyanov V.V. Preobrazhenskii Institut d’Electronique et des Systèmes Montpellier, France E. Tournié F. Gonzalez-Posada G. Boissier Russian Science Foundation CNRS (LIA TeraMIR) ANR (Dirac3D project) French Embassy in Moscow (Vernadski scholarship) LNCMI-CNRS, a member of EMFL ERC project MOMB (No. 320590)
- 69. Author’s publications 69 A1. Bovkun, L.S., et al., Exchange enhancement of the electron g-factor in a two-dimensional semimetal in HgTe quantum wells. Semiconductors, 2015. 49(12): p. 1627-1633. A2. Bovkun, L.S., et al., Magnetospectroscopy of double HgTe/CdHgTe quantum wells. Semiconductors, 2016. 50(11): p. 1532-1538. A3. Bovkun, L.S., et al., Activation conductivity in HgTe/CdHgTe quantum wells at integer landau level filling factors: role of the random potential. Semiconductors, 2017. 51(12): p. 1562-1570. A4. Ikonnikov, A.V., et al., On the Band Spectrum in p-type HgTe/CdHgTe heterostructures and its transformation under temperature variation. Semiconductors, 2017. 51(12): p. 1531-1536. A5. Krishtopenko, S.S., et al., Cyclotron resonance of Dirac fermions in InAs/GaSb/InAs quantum wells. Semiconductors, 2017. 51(1): p. 38-42. A6. Ruffenach, S., et al., Magnetoabsorption of Dirac Fermions in InAs/GaSb/InAs “Three-Layer” Gapless Quantum Wells. JETP Letters, 2017. 106(11): p. 727-732. A7. Bovkun, L.S., et al., Landau level spectroscopy of valence bands in HgTe quantum wells: Effects of symmetry lowering. https://arxiv.org/abs/1711.08783, 2019 (with referees). A8. Bovkun, L.S., et al., Magnetooptics of HgTe/CdTe Quantum Wells with Giant Rashba Splitting in Magnetic Fields up to 34 T. Semiconductors, 2018. 52(11): p. 1386-1391. A9. Bovkun, L.S., et al, Polarization-Sensitive Fourier-Transform Spectroscopy of HgTe/CdHgTe Quantum Wells in the Far Infrared Range in a Magnetic Field. JETP Letters, 2018. 108(5): p. 329-334.
- 70. 70 CdTe HgTe CdxHg1-xTe NostrainCompressiveTensile Band insulator (normal band structure) Gapless material (massless Kane Fermions) Gapless material (inverted band structure) Gapless material (inverted band structure) Inverted band structure with band gap (3D TI) Normal band structure Dirac cone Semimetal Inverted band structure (2D TI) towards bulk Weyl system: two Dirac cones at k≠0 dc dsm THz detectors THz sources X-ray detectors xc Applications Gated structures Doping Double quantum wells 1D-dimensional systems Superlattices … Rogalski, A. (2010). Infrared Detectors. Boca Raton: CRC Press Onshage, A.C., Handbook of Sensors and Actuators. 1997, Elsevier Science B.V.
- 71. 40 50 60 70 80 90 100 110 120 -100 -80 -60 -40 -20 0 20 40 60 Энергияприk=0,мэВ Ширина квантовой ямы, Å 1-я подзона зоны проводимости 1-я валентная подзона 2-я валентная подзона 3-я валентная подзона E1 H1 H2 H3 QW width, A Energy,meV 1st conduction subband 1st valence subband 2nd valence subband 3rd valence subband Width of single QW matters Band insulator (normal band structure) Dirac semimetal Semimetal 2D topological insulator towards bulk Quantum well HgTe width control
- 72. Electronic states in HgTe/CdTe QWs Band insulator d=dc d>dcd<dc d>dsm Dirac cone Topological insulator Semimetal 2D TI (Th): B.A. Bernevig et al. Science 314, 1757 (2006). 2D TI (Exp): M. König et al. Science 318, 766 (2007). SM (Exp): Z. Kvon et al. JETP Lett. 87, 502 (2008).
- 73. cci , hhhhi , lhlhi , shshi , Burt, M.G., The justification for applying the effective-mass approximation to microstructures. Journal of Physics: Condensed Matter, 1992. 4(32): p. 6651. Envelope function approximation Г6 Г8 Г7 Landau level indices: n = −2, −1, 0, …
- 74. 8x8 Hamiltonian (Novik, 2005) Accurate solution for finite amount of bands: DeformationNonspherical (cubic) symmetry terms Hartree potential (SIA) 𝐻 = 𝐻8∙8 + 𝐻ε + 𝑉 𝑧 + 𝐻𝑒𝑥 Novik, E.G., A. Pfeuffer-Jeschke, T. Jungwirth, V. Latussek, C.R. Becker, G. Landwehr, H. Buhmann, and L.W. Molenkamp, Band structure of semimagnetic Hg(1−y)Mn(y)Te quantum wells. Physical Review B, 2005. 72(3): p. 035321.
- 75. c c c c hh lh sh hh lh sh lh hh lh hh sh sh T Pk 2 1 T zPk 3 2 Pk 6 1 zPk 3 2 Pk 6 1 Pk 2 1 Pk 3 1 Pk 3 1 zPk 3 1 zPk 3 1 VU VU VU VU U U R R * S * S V2 V2 R2 R2 S 2 1 * 2 1 S * 3 2 S S 3 2 Pk 2 1 zPk 3 2 zPk 3 2 V2 V2 Pk 6 1 zPk 3 1 Pk 3 1 Pk 3 1 zPk 3 1 Pk 2 1 S * R * R S S 2 1 * 2R S 3 2 * 3 2 S * 2R * 2 1 S Pk 6 1 Kane Hamiltonian
- 77. Phase transition at critical field symmetry broken symmetry (011) (111) (001) (013) No BIA induced interaction of Landau levels -2 and 0 should be observed for such structures BIA induced interaction of Landau levels -2 and 0 was observed for such structures 77
- 78. 0 2 4 6 8 10 -60 -40 -20 0 20 40 60 10 -1 1 -2 Energy(meV) Magnetic field (T) 0 α β γ α β γ splitting 78 Avoided crossing in HgTe QWs
- 79. H ≠ 0 n = 3 n = 2 n = 1 n = 0 79 Electric dipolar selection rules
- 81. k m k kE *2 )( 22 Bychkov, Y.A. and E.I. Rashba, JETP Letters, 1984. 39(2): p. 78. Spin-orbit coupling Asymmetric confinement potential Spin splitting Bychkov-Rashba spin-splitting Linear terms presented in Kane model as well
- 82. Расщепление линии циклотронного резонанса К.Е.Спирин, А.В.Иконников, А.А.Ластовкин, В.И.Гавриленко, С.А.Дворецкий, Н.Н.Михайлов. Спиновое расщепление в гетероструктурах HgTe/CdHgTe (013) с квантовыми ямами Письма в ЖЭТФ. 92, 65 (2010)
- 83. План доклада 83 o Исследование одиночных квантовых ям HgTe/CdHgTe • Замешивание электронных состояний в валентной зоне симметричных КЯ • Исследование эффектов структурной асимметрии в зоне проводимости • Активационная проводимость в квантовых ямах HgTe/CdHgTe при целочисленных факторах заполнения уровней Ландау: роль случайного потенциала (по структуре диссертации)
- 84. Типичные зависимости Rxx и Rxy при разных температурах Бовкун, Л.С., et al., Активационная проводимость в квантовых ямах HgTe/CdHgTe при целочисленных факторах заполнения уровней Ландау: роль случайного потенциала. ФТП, 2017. 51(12): p. 1621.
- 85. Определение эффективной массы и профиля уширения УЛ Бовкун, Л.С., et al., Обменное усиление g-фактора электронов в двумерном полуметалле в квантовых ямах HgTe. ФТП, 2015. 49(12): p. 1676-1682. m*, τq, ΔT, g*
- 86. Определение щели подвижности ΔT Бовкун, Л.С., et al., Активационная проводимость в квантовых ямах HgTe/CdHgTe при целочисленных факторах заполнения уровней Ландау: роль случайного потенциала. ФТП, 2017. 51(12): p. 1621. m*, τq, ΔT, g*
- 87. Определение энергетических щелей в магнитном поле Бовкун, Л.С., et al., Активационная проводимость в квантовых ямах HgTe/CdHgTe при целочисленных факторах заполнения уровней Ландау: роль случайного потенциала. ФТП, 2017. 51(12): p. 1621. m*, τq, ΔT, g* Магнитооптические переходы, дающие напрямую g* запрещены правилами отбора
- 88. 88 Обоснован формализм Гаусса уширения УЛ
- 89. 89
- 90. Положения, выносимые на защиту 90 В квантовых ямах HgTe/CdHgTe эффекты обменного усиления спинового расщепления уровней Ландау несущественны, и определяемые из магнитотранспорта величины спиновых щелей зоны проводимости находятся в хорошем согласии с результатами одноэлектронных расчетов. 3
- 91. E1∩H1 E1∩H2 E2∩H1 E2∩H2 Band Insulator 1. Normal band QW + Normal band QW --> Inverted Band Ordering. A) E1∩H1 = massive fermions B) E1∩H2 = Dirac cone (1 pair) C) E2∩H2 = Dirac cone (2 pair) 2. Available phases A) Band insulator (white) B) Topological insulator (I) C) Metallic “bilayer graphene” with gap opening at perpendicular field (blue) D) Band insulator with 2 pairs of helical edge states E) Semimetal (SM) Summary on phase diagram
- 92. 92 Studies of double quantum wells In sample (A) with normal dispersion: • Fermi level could be modified with illumination, revealing band asymmetry • Energy splitting (E2-E1) > (H2-H1) due to the different transparency of the barrier for Г6 and Г8 states Doubling of interband transitions observed in both samples, allowing estimation of interlayer coupling value In sample (B) with “bilayer graphene-like” dispersion: • Interlayer coupling value is comparable with cyclotron energy for B < 2 T, decoupling of QWs observed in higher fields • Opening of fundamental gap was observed due to the presence of build-in electric field
- 93. План доклада 93 o Исследование одиночных квантовых ям HgTe/CdHgTe o Исследование двойных квантовых ям HgTe/CdHgTe o Дираковские фермионы в InAs/GaSb/InAs • Замешивание электронных состояний в валентной зоне симметричных КЯ • Исследование эффектов структурной асимметрии в зоне проводимости • Активационная проводимость в квантовых ямах HgTe/CdHgTe при целочисленных факторах заполнения уровней Ландау: роль случайного потенциала (по структуре диссертации)
- 94. 94 Трехслойные структуры InAs/GaSb/InAs Liu, C., et al., Quantum Spin Hall Effect in Inverted Type-II Semiconductors. Physical Review Letters, 2008. 100(23): p. 236601. Krishtopenko, S.S. and F. Teppe, Quantum spin Hall insulator with a large bandgap, Dirac fermions, and bilayer graphene analog. Science Advances, 2018. 4(4).
- 95. 95 Коническая и параболические подзоны Krishtopenko, S.S., et al., Cyclotron resonance of dirac fermions in InAs/GaSb/InAs quantum wells. Semiconductors, 2017. 51(1): p. 38-42. E2 E1 «конус»
- 96. 96 Трехслойные структуры InAs/GaSb/InAs Ruffenach, S., et al., Magnetoabsorption of Dirac Fermions in InAs/GaSb/InAs “Three-Layer” Gapless Quantum Wells. JETP Letters, 2017. 106(11): p. 727-732.
- 97. Положения, выносимые на защиту 97 В спектрах циклотронного резонанса электронов в трехслойных симметричных квантовых ямах InAs/GaSb/InAs, ограниченных барьерами AlSb с расчетными толщинами слоев, соответствующих бесщелевой зонной структуре с дираковским конусом в центре зоны Бриллюэна, в классических магнитных полях наблюдается уширение и сдвиг линии ЦР при уменьшении концентрации электронов, свидетельствующие о наличии «конической» и параболической подзон в зоне проводимости. В квантующих (до 34 Тл) магнитных полях наличие «конической» подзоны проявляется в возникновении новой линии поглощения, обусловленной переходом с нижнего уровня Ландау. 5
Editor's Notes
- Good afternoon dear colleagues and jury members, I would like to thank everyone for coming and showing interest towards my talk. I also would like to thank jury members for reading my manuscript and providing valuable comments. My work is dedicated to band structure of narrow-band semiconductor systems. Thesis was supervised by Marek Poremski, …
- Here’s the outline from my talk. It consists of introduction of physical properties and band-engineerin possibilities of systems under consideration, followed by a short review of theoretical approach of describing band structure. Before presenting original part of the work, I will briefly explain experimental techniques and principles.
- Mercury cadmium telluride is semiconductor with zinc-blende structure, as many other materials, for example GaAs. That is why understanding of band structure of this particular material reflecting a wide topic of semiconductor physics and narrow-band semiconductors especially. Mercury cadmium telluride is a ternary compound composed of two self-standing materials: CdTe and HgTe. And as you can see from the slide, there is a dramatic difference between their bulk structure. As GaAs or Si, CdTe possess normal band structure where conduction band is formed of s-type states represented by point group Г6, while conduction band is formed out of point group Г8. In HgTe it is vice versa, and this particular band ordering, called inverted band structure is one of the key requirements to obtain topological insulators. The reason for the inversion are relativistic corrections to the Hamiltonian, which are particularly pronounced in this material, due to the heavy Hg atoms [100].
- In 1959, a publication by Lawson et al [1] triggered the development of variable band gap Hg1−xCdxTe (HgCdTe) alloys providing an unprecedented degree of freedom in infrared (IR) detector design. The idea is rather simple – is to replace Cd atoms with Hg, slowly decreasing the value of the bandgap. At specific ratio of 17% one will end up with gapless band structure, consisting of two touching cones and one massive band at Г-point.
- As well as in the bulk material, one can control bandgap not only with composition of the compound, but with thickness of quantum well. In order to obtain quantum confinement one may put layer of gapless HgTe between CdTe with opened gap. For narrow quantum well the influence of interfaces is barrier band structure dominates over natural. Thus we obtain normal band structure depicted at the bottom of the screen.
- If one increase the thickness of quantum well then the natural properties of HgTe eventually result in closing band gap, forming Dirac cone dispersion law at critical thickness. Afterwards, gap will reopen, and the band structure became inverted, with mixing on s and p states in the vicinity of Г-point.
- The existence of edge states in HgTe quantum well were predicted theoretically and soon manifested in observation of quantum spin Hall effect. In the bulk, there is a finite bandgap, and system behave as an insulator. At the edges, electron-like subband is moving toward higher energies and the Dirac cone appears at the intersection. It was shown in the experiment that electrons with different spin can travel along the edge of the sample without any dissipation due to time-reversal symmetry, forbidding scattering in such a system.
- At this point I would like to illustrate the similarity between band structure engineering by the mean of bulk composition and QW thickness. If CdTe plays major role in the material, then we end up with normal band structure. In the contrary, HgTe is likely to form gapless systems with inverted band structure – important property for topological matter. The existence of the additional massive band doesn’t allow bulk HgTe to demonstrate properties of topological insulator, as long as there is no gap opens in the bulk.
- It was recently predicted that one can actually tune the band structure in the system with choosing a proper substrate. On one hand, it is possible to open a band gap with tensile strain to reveal topological properties of 3D TI. On the other hand, compressive strain may eventually create the Weyl system.
- One may also create gated structures and introduce some doping in order to control Fermi level in the system. Another direction is to increase number of quantum wells in the structure. All of this became possible due to technological progress of growing of mct-heterostructures, stimulated by particular interest of creating devices working in THz range.
- In the introduction I’ve tried to convince you how versatile is the band structure of the chosen heterosystem. Nowadays, people are trying to make more and more sophisticated devices, where understanding of the band structure is facing different challenges. And some of them I will discuss later…
- In the following slides I am going to describe theoretical approach of calculating band structure.
- In 1957, Evan Kane proposed a theoretical approach to describe band structure of indium antimonide. This is A3B5 semiconductor with narrow gap and zinc blende crystal lattice. It was proposed to calculate energy spectra of the system in a kp theory with an accurate treatment of conduction and valence band interactions while higher bands are treated by perturbation theory.
- The Kane model was applied to describe MCT. The set of Bloch functions was chosen following the symmetry of the system in Г-point. Conduction subband, light and heavy holes subband plus splitted by spin-orbit interaction valence subband are considered exactly. The material and model parameters presented in the work of Novik allows to adequately interpret experimental results, for example non-parabolicity of conduction band.
- In quasi-2D quantum wells (QWs) and heterostructures one should also consider effects of reduced symmetry. It is well-known that in the presence of asymmetrical electrostatic potential the spin degeneracy is removed and one can observe so called Bychkov-Rashba splitting. Secondly, there is no inversion center in the crystals with zinc blende structure, and addition bulk inversion asymmetry term arises. A third contribution to zero-field spin-splitting can be the low microscopic symmetry of the atoms at an interface.
- In my talk I will address Kane model with axial symmetry, and impact of additional terms I will call as extended model. Majority of experimental papers use either single Kane Hamiltonian or take into account electrostatic potential for gated structures. The interpretation of experiments we rather adequate so far. Nevertheless, I would like to show two particular experimental cases, where samples under studies have no asymmetric potential, but display effects of reduced symmetry in the system.
- On this particular slide presented calculations of Landau levels in two samples with inverted band structure. I will talk about Landau levels in more detail afterwards, but now I would like to focus your attention to the very particular point – crossing of two Landau levels in the axial model. In the model with additional term, taking into account bulk inversion asymmetry – the gap is opening due to interaction between Landau levels.
- The left color plot depicts avoided crossing of Landau levels in the magneto-optical experiment, corresponding to gap opening. So this is the first evidence of reduced symmetry in the system.
- On this slide presented calculations of the band structure in the symmetrical sample, where effect of electrostatic potential is negligible. Magnetotransport studies presented in cited papers allowed to reconstruct two spin branches in the valence band and explain the results taking into account effects of interface inversion asymmetry.
- In the past slides I’ve introduced the Hamiltonian that is used to describe band structure in samples under study. I also demonstrate particular experiments, where use of axial model cannot interpret obtained result. The takeaway message from this section is that extended model is needed in case of induced electrostatic potential, when your study p-type samples and in case of narrow heterostructures, where effect of interface asymmetry has noticeable impact.
- Motivation of my work is to find out which effect is stronger. In order to reach my goal we will follow the step presented on the slide Conduct experimental studies of…
- Before presenting original part of the work I would like to describe experimental details and principles. As we are interested in narrow gap semiconductor system with small effective masses, characteristic energy of optical response belongs to far-infrared region. Additionally, HgTe QW structures are sufficiently transparent in the far-infrared region, so the study of transmission spectra and polarization experiments could provide valuable information about band structure.
- Cyclotron resonance is an example of magneto-optical experiments. Electrons in the plane perpendicular to magnetic field starts to move along the orbits due to the Lorentz force. Cyclotron frequency of such a motion is proportional to the value of magnetic field and depends on effective mass. Orbital motion of charged particles creates modulation of electric field and may resonantly absorb incoming light if scattering allows. To fulfill the required condition one should either study good-quality samples at low temperatures or go to higher magnetic fields (which are located on the ground floor of this building). In classical picture there is no energy quantization, so electrons can have orbits of any size. This fact contradicts to experimental results, so we need quantum mechanics.
- For a two-dimensional parabolic band in magnetic fields we obtain set of Landau levels, separated by the value of energy hwc, as one can see from this equation. Optical transitions with resonant frequency allowed only between landau levels with index difference of plus or minus one, where sign defines the polarization of adsorbed light.
- In real structures (and especially in gapless semiconductors) we can have non-parabolic conduction band, which gives rise to non-equidistant Landau levels. In this case, energy of intraband transitions might be different for a fixed magnetic field, so it can be resolved in the experiment, providing information about band structure and Fermi energy
- In real crystal we have conduction and valence band . Each band gives a fan of Landau levels. Thus, we can observe interband transitions, as beta and alpha-minus. In our theoretical model selection rules are the same for both types of transitions. It is worth to mention, that in gapless and narrowband structures, energies of CR and interband transitions might be of the same range, as it seen from the picture.
- With increase of magnetic field not only the spacing between Landau level increasing, but also degeneracy for each of them, so less and less Landau levels are occupied. Due to the occupation effect, we cannot observe transitions from empty levels, as well as transitions to fully occupied levels. The intensity of the line is proportional to amount of carries absorbing the light.
- To summarize, observation of magnetooptical transitions can provide information concerning effective mass and bandgap of a system. Magnetooptical picture obtained for a broad range of magnetic field coupled with occupational effect allow to reconstruct Landau levels, directly linked with a band structure.
- In order to perform magneto-optical experiments a system of coupled Fourier-spectrometer and magnet is used. Radiation from source is passing through beamsplitter and travel inside the light-pipe towards sample. Transmitted light is detected with a bolometer, located right after the sample, and send to computer after amplification.
- Sample and reference is mounted on top of rotational holder, so it is possible to perform absolute measurements of transmission. The holder is located here and then one may see attached bolometer. The light from spectrometer travel in parallel beam and entering the probe, located inside the cryostat with superconducting magnet.
- We also perform measurement of magnetotransport in 4-terminal Van der Pauw configuration to determine type of conductivity and carrier density in the sample. It is possible to modify carrier density with illumination of LED, located nearby the sample. In order to perform polarization-sensitive spectroscopy we use combination of fixed and rotating linear polarizers to study effect of Faraday rotation.
- In the following sections I will present original part of the work, starting from the studies of p-type samples.
- The samples were grown in Novosibirsk by the group of N. Mikhailov and S. Dvoretsky with MBE setup. GaAs was used as substrate, following thick layer of CdTe to remove the lattice mismatch, in other words not to introduce any strain. The pure HgTe quantum well is located between barriers, and protective layer is covering the structure. The samples of different width were studied, covering the whole range of normal, near gapless and inverted band structure.
- Here I present two typical magnetotransport measurements for samples with normal band structure. Longitudinal resistivity shown in black and follows left axis, while transverse resistivity is red. Initial magnetotransport characterization confirms hole-type conductivity in all the samples with relatively low density. From positions of Shubnikov de Haas oscillations we were able to estimate position of Fermi level in the system and account for occupational effect.
- At the left part of the slide presented experimental data obtained for the almost gapless sample. All the spectra corresponds to relative transmittance, in other words, normalized by zero-field spectra of the sample, opaque regions are masked with grey. On the left side solid lines are Landau levels and arrows denote observed transitions. The Fermi level is shown with dotted line. At 10 Tesla all Landau levels in the valence band are occupied and all Landau levels in conduction band are empty. There are two interband transitions observed, one of them is beta transition well known in the literature for studies of n-type samples. In lower magnetic field we additionally observe intraband transition in the valence band denoted as beta-minus. I’d like to notice that these transitions are characteristic for p-type samples only and may be used as reference in future studies.
- At this slide experimental data is depicted with different symbols, while calculation in axial model is show with lines. Since all of samples under consideration have relatively small density of holes, high field spectra contain mentioned earlier interband transitions alpha-minus and beta. Extrapolation of transition energy toward zero magnetic field allow to estimate energy gap of the sample. With increase of QW width the energy gap closes and then opens up again for sample with the inverted band structure. At the low magnetic fields highest Landau levels in the valence band are depopulated, allowing observation of intraband transitions beta-minus and delta-minus.
- Overall magnetooptical experiment could be adequately explain with calculations in axial model, except highlighted regions, not only for intraband, but for interband transitions. The most pronounced effect we will discuss for the sample with QW thickness of 5 nm.
- Rectangular area is presented on the left side as color plot and the raw data for 2 Tesla is show as well. Both beta minus asterisk and alpha minus asterisk lines are present for the wide range of magnetic field up to 4 Tesla. None of this lines could be explained in the axial model.
- Effects of reduced symmetry in extended model allow closely spaced Landau levels in valence band to interact, resulting in non-zero matrix element for axially forbidden transitions. Indeed, from our calculations we find out, that forbidden alpha transition is initiated from lower in energy level than original one and that is why, energy of forbidden transition is higher. The same logics applied for forbidden beta minus transition. And on top of that, appearance of the lines matches well with occupational effect.
- In the sample with inverted band structure the experimental results are not consistent with axial model as well. Namely, transition beta-minus and beta deviates from their axial trajectories, but follows calculations of extended model.
- In this slide we compare magneto-optical response of n-type and p-type samples. In both cases transition beta is present in high-field region and approaching anticrossing with similar trajectory. In low field, transition alpha observed for n-type sample, while for p-type sample we find interband transition beta-minus, dramatically deviating from axial calculations.
- As we just witnessed effect of natural asymmetry effect in the valence band, we decided to study effects of induced asymmetry in the conduction band, to observe it in a more clear conditions.
- The samples were grown in the same manner, but asymmetry was provided by means of selective doping of the bottom barrier. In this particular section we will compare results for samples with inverted and normal band structures.
- In the previous works initially symmetrical samples were studied, where asymmetrical potential was induced by voltage applied to the gate on top of the structure. One of particular outcome of this studies is observation of dramatically different splitting between first and second conduction band. The splitting of the lower band is shown with this line and the value is proportional to cubic power of wavevector. The splitting in the upper band is much less.
- This difference was explained as internal property of electron-like states and hole-like states in the work of Winkler. Since above-mentioned experiments we performed on structures with inverted band structure, effect of splitting in conduction band was explained as admixture of p-states in the first conduction band. In turn, one can assume, that in normal bans structure, there should be relatively small splitting.
- Since this work is dedicated to doped samples we observe large electron density: high-frequency Shubnikov de Haas oscillations starting below 1 Tesla at 4.2K. The beating patterns of SdH oscillations is the clear attribute of multiple carrier system. Fourier analysis of the data allow to extract values of carrier densities in each subband, an we have clearly observed spin splitting in first subband. From carrier density we can calculate values of Fermi vector and evaluate Fermi energy in our system.
- Splitting of the cyclotron resonance line were observed in the low fields as well, and we extract values of effective mass from linear dependence of cyclotron energy versus magnetic field. In higher magnetic field quasi-classical regime is no longer applied and one should considered different transitions between landau levels. We have also calculated effective masses from dispersion laws following quasi-classical formula, presented on the slide, where A stands for area of specific branch. Calculated values of effective mass predict smaller splitting than observed experimentally.
- Solving Poisson equation with input parameters of carrier concentration from transport measurements we reconstruct confinement potential for our samples. Effects of bulk asymmetry and interface asymmetry were also considered.
- Here I present results of calculation, where conduction bands have different splitting. Our calculations presented on this slide, revealing different splitting in the first and second conduction subbands. First conduction band is splitted mainly due to Bychkov-Rashba effect with negligible participation from other asymmetries. For the second conduction band, on the contrary, inversion and interface asymmetries prevail
- Here we present distribution of electron-like Г6 and hole-like Г8 wave function in the direction of growth. In fact, there is a noticeable admixture of Г8 states in conduction band even for the sample with the normal band structure. That is why the value of the splitting is still noticeable in experiments.
- In the top left corner presented the scheme samples under study, which consist of two identical quantum wells of thickness d, separated by the wide gap barrier of thickness t. For different combination of d and t one can obtain convenient phases of topological and band insulator as well as semimetal for large QW width. The same phases are present in single quantum wells of different thickness. If we consider relatively thick barrier, then everything below critical thickness correspond to normal band structure, then we have inverted and semimetallic band structure. There is the specific gapless band structure of two touching at the gamma point parabolas, resembling bilayer graphene dispersion law. The designed parameters of samples (and in brackets – adapted from experimental results) presented on the slide.
- In the presence of transparent barrier ground states of each single quantum wells can form symmetric and anti-symmetric wave-functions, resulting in appearance of two distinct energies for bonding and antibonding states. Similar picture appears when you have two tunnel-coupled QWs with normal band structure. As result both conduction and valence band form bonding and anti-bonding bands, as one can see from the calculations presented on the slide. Let me show the experimental results obtained for this sample.
- At the bottom of the page presented the color plot of absorption, where absorption features are shown in yellow, green and blue in order of increase the effect. Initially, the Fermi level of the sample are located in the valence band with magnetooptical response shown here. Illumination of the sample with LED introduce additional electrons from surrounding barriers. Thus, we were able to lift up the Fermi energy to the conduction band. In this particular case we observe strong absorption line within conduction band. From the slope of cyclotron resonance lines one can obtain values of effective mass with classical formula. It is seen, that the valence band is occupied with heavier particles.
- The transparency of the barrier depends from the effective mass as shown is the formula. Therefore, light particles tunnels more efficient and it leads to pronounced appearance of bonding and anti-bonding band.
- Intraband transitions previously discussed dominates magneto-optical response in low energy region, where alpha-minus transition is well-known from literature, but beta and delta-minus are exactly the same transitions that we observe in single quantum wells of p-type. In the high energy part of the experiment we observe additional interband lines, presented on the chart as symbols. Well known for intraband line beta is now doubled, because now we have transition to bonding band E1 and anti-bonding band E2.
- If we overlay calculations of transition energy as lines one may notice, that both lines goes parallel in magnetic field. If we track beta-one transition down to low magnetic field we obtain the value of fundamental gap. The same procedure, but with beta-two line gives the energy difference between conduction subband
- Now, if we couple two Dirac-cone samples we will obtain specific band structure. It is gapless and at the Г-point two parabolas come into contact (adjoin). In the contrast with previous sample, not only anti-bonding conduction band are separated in energy, but bonding valence band also have some offset.
- So, our old friends beta1 and beta2 lines are clearly visible in the spectra, going parallel to each other. Beta1 is now intraband transition and should follow to zero energy, beta2 in interband. The energy difference between this transition allow to extract one important parameter of the band structure, Gamma 1, so called interlayer coupling, coming from physics of bilayer graphene.
- Namely gamma1 is presented in Hamiltonian of bilayer graphene. The broad view of tight binding calculations shown on the top, and the closer vicinity of the K-point is enlarged. Band structure of tunnel coupled double quantum well resembles well the structure of bilayer graphene. There are two important differences. Bilayer graphene have such dispersion both in K and K’ points, while sample under study – at Г-point. There is a striking difference between values of interlayer coupling, 20 meV is extremely small energetic parameter for our system.
- In the sample under consideration, the lowest transition corresponds to cyclotron resonance in the valence band. It is seen from the graph, that energy of cyclotron resonance is comparable to the value of interband coupling already in the magnetic field of 2 Tesla. This means, that for higher magnetic fields, our quantum wells are effectively decoupled. Each of the conduction band should demonstrate Dirac cone behaviour, where the splitting of Landau levels increases as square root of B. In order to decouple bilayer graphene one need magnetic fields higher than 100 Tesla.
- Detailed analysis of lines beta1 and beta2 reveal additional splitting both in high fields and in low fields. The final Landau levels is bonding and antibonding states in conduction band and have no degeneracy. Initial level with index -2 is double degenerated, because it exists in each of quantum well. Our calculation in the presence of built-in electrostatic potential support obvious explanation of this splitting. At the bottom of the page we compare electron states between single and double quantum wells. In the absence of electrical field we have resonant coupling and twofold degeneracy of each level. Additional electric field lifts this degeneracy and brings the wells out of resonance, electrons tunnels toward lower potential energy and enhance energy splitting. The same potential have much smaller effect in case of single quantum well.
- In conclusions I’d like to answer to the question: when asymmetry effect are relevant? Valence band in symmetrical quantum wells are sensitive to both bulk inversion and interface inversion asymmetries. We observe axially forbidden transitions at low magnetic field and additionally confirm avoided crossing in p-type samples. Conduction band in asymmetric quantum wells is sensitive to electrostatic potential not only in samples with inverted band structure, but in samples with normal band ordering due to admixture of p-type states. Magneto-optical experiments on double quantum wells confirm tunnel transparency of the barrier. In turn, mct mercury systems can host additional phase of bilayer graphene. Distinctive difference of our system is small value of interlayer coupling. Additionally, we show that double quantum wells are sensitive to electrostatic potential, and opening of fundamental gap was shown experimentally.
- At the end of my talk I would like to acknowledge my colleagues and colaborators both in Russia and in France. Thank you for attention.
- MCT in its bulk form are irreplaceable material so far for creating optoelectronic devices operating in terahertz range. The dispersion law of electrons in such structures is nonparabolic and similar to the one of relativistic particles. Usually it is called Dirac dispersion law. The band gap can be even negative which results in special state called topological insulator.
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- Let’s fix quantum well width at 7.5 nm. For the 2nm barrier we will find familiar bilayer graphene metallic phase. There’s one more special line that we didn’t discuss yet which corresponds to crossing og E2 electron-like zone with HH1 and HH2. At this point another dirac cone occurs